一类非光滑规划问题的最优性和对偶

研究一类非光滑多目标规划问题,给出了该规划问题的三个最优性充分条件.同时,研究了该问题的对偶问题,给出了相应的弱对偶定理和强对偶定理.     

Symmetric duality for a class of nonlinear fractional programming problems

A pair of symmetric dual nonlinear fractional programming problems is presented and duality theorems are established under pseudoinvexity-pseudoincavity (and invexity-incavity, respectively) type assumptions on the kernel function. Special cases are particularly discussed to show that this paper extends some work appeared in this area.     

Symmetric duality for a class of multiobjective fractional programming problems

This paper is concerned with symmetric duality for a class of nondifferentiable multiobjective fractional programming problems. Two weak duality theorems and two strong duality theorems are proved. Discussion on some special cases shows that results in this paper extend previous work in this area.     

On sufficiency and duality for a class of interval-valued programming problems

In this paper, we are concerned with an interval-valued programming problem. Sufficient optimality conditions are established under generalized convex functions for a feasible solution to be an efficient solution. Appropriate duality theorems for Mond-Weir and Wolfe type duals are discussed in order to relate the efficient solutions of primal and dual programs.     

一类多目标广义分式规划问题的最优性条件和对偶

研究了一类不可微多目标广义分式规划问题．首先,在广义Abadie约束品性条件下,给出了其真有效解的Kuhn—Tucker型必要条件．随后,在（C,a,P,d）一凸性假设下给出其真有效解的充分条件．最后,在此基础上建立了一种对偶模型,证明了对偶定理．得到的结果改进了相关文献中的相应结论．     

Constructing a perfect duality in infinite programming

One version of the infinite Farkas Lemma states the equivalence of two conditions, (1)yb 0 wheneverya ^j 0 forj=1,2,.. and (2)b cl C, whereb and alla ^j are inR ⁿ andC is the convex cone spanned by all thea ^j 's. In this paper an ascent vector specifies a direction along which an arbitrarily small movement fromb with enterC. A Fredholm type theorem of the alternative characterizes the set of all ascent vectors associated with an arbitrary system of linear inhomogeneous inequalities in a finite number of variables. As a consequence, a pair of infinite programs is constructed which is in perfect duality in the sense that (p1) if one program is consistent and has finite value, then the other is consistent and (p2) if both programs are consistent, then they have the same finite value. The duality is sharp in that the set of all feasible perturbations along rays is determined.This research was supported by NSF Grants GK-31833 and ENG76-05191. The paper is a revision of an earlier report of June 1975.     

Weak stability and strong duality of a class of nonconvex infinite programs via augmented Lagrangian

In this paper we deal with weak stability and duality of a class of nonconvex infinite programs via augmented Lagrangian. Firstly, we study a concept of weak-subdifferential of an extended real valued function on a topological linear space. Augmented Lagrangian functions and a concept of weak-stability are constructed. Next, relations between weak-stability and strong duality of problems via augmented Lagrangians are investigated. Applications for convex infinite programs are discussed. Saddle point theorems are established. An illustrative example is given.     

A class of infinite dimensional linear programming problems

We study the infinite dimensional linear programming problem. The previous work done on this subject defined the dual problem in a small space and derived duality results for such pairs of problems. But because of that and of the strong requirements on the functions involved, those theorems do not actually hold in many applications. With our formulation, we define the dual problem in a larger space and obtain new duality results under, generally, mild assumptions. Furthermore, the solutions turn out to be extreme points of the unbounded, but w¹-locally compact, feasibility set. For this purpose, we did not try a constructive proof of our duality results, but instead we examine the problem from a more abstract point of view and derive results using general ideas from the theory of convex analysis in normed spaces [R. T. Rockafellar, “Conjugate Duality and Optimization,” SIAM, Philadelphia, Penn., 1973, and R. Holmes, “Geometric Functional Analysis,” Springer-Verlag, New York, 1975]. Our work extends previous results in this area, which appeared in [N. Levinson, J. Math. Anal. Appl.16 (1965) 73–83, and W. Tundall, SIAM J. Appl. Math.13 (1965), 644–666].     

A duality theorem for a special class of programming problems in complex space

Duality relations for the programming problem of a special class where the objective function is a sum of positive-semidefinite quadratic forms, and a sum of square roots of positive-semidefinite quadratic forms, over a convex polyhedral cone in complex space are considered. The duality relations between the primal problem and its dual are established.     

Optimality conditions and duality for a class of nondifferentiable multi-objective fractional programming problems

A class of multi-objective fractional programming problems (MFP) are considered where the involved functions are locally Lipschitz. In order to deduce our main results, we give the definition of the generalized (F,θ,ρ,d)-convex class about the Clarke’s generalized gradient. Under the above generalized convexity assumption, necessary and sufficient conditions for optimality are given. Finally, a dual problem corresponding to (MFP) is formulated, appropriate dual theorems are proved.      

Optimality conditions and duality for a class of nondifferentiable multiobjective generalized fractional programming problems

This paper studies a class of multiobjective generalized fractional programming problems, where the numerators of objective functions are the sum of differentiable function and convex function, while the denominators are the difference of differentiable function and convex function. Under the assumption of Calmness Constraint Qualification the Kuhn-Tucker type necessary conditions for efficient solution are given, and the Kuhn-Tucker type sufficient conditions for efficient solution are presented under the assumptions of （F, α, ρ, d）-V-convexity. Subsequently, the optimality conditions for two kinds of duality models are formulated and duality theorems are proved.     

Lagrangian duality and saddle points for sparse linear programming

The sparse linear programming(SLP) is a linear programming problem equipped with a sparsity constraint, which is nonconvex, discontinuous and generally NP-hard due to the combinatorial property involved.In this paper, by rewriting the sparsity constraint into a disjunctive form, we present an explicit formula of the Lagrangian dual problem for the SLP, in terms of an unconstrained piecewise-linear convex programming problem which admits a strong duality under bi-dual sparsity consistency. Furthermore, we show a saddle point theorem based on the strong duality and analyze two classes of stationary points for the saddle point problem. At last,we extend these results to SLP with the lower bound zero replaced by a certain negative constant.     

Exact augmented Lagrangian duality for mixed integer linear programming

We investigate the augmented Lagrangian dual (ALD) for mixed integer linear programming (MIP) problems. ALD modifies the classical Lagrangian dual by appending a nonlinear penalty function on the violation of the dualized constraints in order to reduce the duality gap. We first provide a primal characterization for ALD for MIPs and prove that ALD is able to asymptotically achieve zero duality gap when the weight on the penalty function is allowed to go to infinity. This provides an alternative characterization and proof of a recent result in Boland and Eberhard (Math Program 150(2):491–509, 2015, Proposition 3). We further show that, under some mild conditions, ALD using any norm as the augmenting function is able to close the duality gap of an MIP with a finite penalty coefficient. This generalizes the result in Boland and Eberhard (2015, Corollary 1) from pure integer programming problems with bounded feasible region to general MIPs. We also present an example where ALD with a quadratic augmenting function is not able to close the duality gap for any finite penalty coefficient.     

Strong duality for infinite-dimensional vector-valued programming problems

LetX,Y andZ be locally convex real topological vector spaces,A?X a convex subset, and letC?Y,E?Z be cones. Letf:X→Z beE-concave andg:X→Y beC-concave functions. We consider a concave programming problem with respect to an abstract cone and its strong dual problem as follows: $$\begin{gathered} (P)maximize f(x), subject to x \in A, g(x) \in C, \hfill \\ (SD)minimize \left\{ {\mathop \cup \limits_{\varphi \in C^ + } \max \{ (f + \varphi \circ g)(A):E\} } \right\}, \hfill \\ \end{gathered} $$ , whereC ⁺ denotes the set of all nonnegative continuous linear operators fromY toZ and (SD) is the strong dual problem to (P). In this paper, the authors find a necessary condition of strong saddle point for Problem (P) and establish the strong duality relationships between Problems (P) and (SD).     

Symmetric duality for lexicographic linear programming problems

A symmetric pair of lexicographic linear-programming (LP) problems, connected by regular relations, is formulated for problems of multicriteria linear optimization. A duality theorem for improper linear-programming problems (ILPP) is constructed in terms of lexicographic optimization.Translated from Ukrainskii Maternaticheskii Zhurnal, Vol. 44, No. 6, pp. 766–773, June, 1992.     

Optimality conditions and Lagrangian duality in continuous-time nonlinear programming

Using a perturbation approach, the Kuhn-Tucker saddlepoint and stationary-point optimality conditions and a Lagrangian duality theory are established for a general class of continuous-time nonlinear programming problems. It is shown that most of the duality formulations in the existing literature of continuous programming are special cases of this Lagrangian formulation.     

Stability and augmented Lagrangian duality in nonconvex semi-infinite programming

In this paper the pseudo-Lipschitz property of the constraint set mapping and the Lipschitz property of the optimal value function of parametric nonconvex semi-infinite optimization problems are obtained under suitable conditions on the limiting subdifferential and the limiting normal cone. Then we derive sufficient conditions for the strong duality of nonconvex semi-infinite optimality problems and a criterion for exact penalty representations via an augmented Lagrangian approach. Examples are given to illustrate the obtained results.     

Algebraic duality theorems for infinite LP problems

In this paper, we consider a primal-dual infinite linear programming problem-pair, i.e. LPs on infinite dimensional spaces with infinitely many constraints. We present two duality theorems for the problem-pair: a weak and a strong duality theorem. We do not assume any topology on the vector spaces, therefore our results are algebraic duality theorems. As an application, we consider transferable utility cooperative games with arbitrarily many players.     

A duality theorem for continuous-time linear programming problems

Summary In dealing with dynamic economic policy models one encounters optimization problems whose objective function is an integral of a linear function of a finite number of continuous variables and whose constraints are linear integral inequalities. A set of intertemporal efficiency conditions (equilibrium conditions) yielding the optimal policy are given. By approximating the continuous problem by a set of discrete problems and appealing to a well known convergence theorem in functional analysis a continuous analog of the duality theorem is proved.

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Zusammenfassung Bei der Beschäftigung mit dynamischen Modellen der ökonomischen Politik stößt man auf Optimierungsprobleme, deren Zielfunktion ein Integral einer linearen Funktion von einer endlichen Anzahl stetiger Variablen ist und deren Beschränkungen lineare Integral-Ungleichungen sind. Eine Menge intertemporaler Effizienz-Bedingungen (Gleichgewichtsbedingungen), die zur optimalen Politik führen, sind gegeben. Durch Approximation des kontinuierlichen Problems mittels einer Menge von diskreten Problemen und Berufung auf einen wohlbekannten Konvergenzsatz aus der Funktionalanalysis wird ein stetiges Analogon des Dualitätstheorems bewiesen.

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The author is indebted to Mr.Arnold Faden for helpful suggestions and to ProfessorKarl A. Fox andGerhard Tintner for encouragement during the preparation of the paper. This research has been partially supported by a grant from the Ford Foundation to the School of Business Administration administered by the Center for Research in Management Science, University of California, Berkeley.

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Vorgel. v.:G. Tintner.     

Strong duality for robust minimax fractional programming problems

We develop a duality theory for minimax fractional programming problems in the face of data uncertainty both in the objective and constraints. Following the framework of robust optimization, we establish strong duality between the robust counterpart of an uncertain minimax convex–concave fractional program, termed as robust minimax fractional program, and the optimistic counterpart of its uncertain conventional dual program, called optimistic dual. In the case of a robust minimax linear fractional program with scenario uncertainty in the numerator of the objective function, we show that the optimistic dual is a simple linear program when the constraint uncertainty is expressed as bounded intervals. We also show that the dual can be reformulated as a second-order cone programming problem when the constraint uncertainty is given by ellipsoids. In these cases, the optimistic dual problems are computationally tractable and their solutions can be validated in polynomial time. We further show that, for robust minimax linear fractional programs with interval uncertainty, the conventional dual of its robust counterpart and the optimistic dual are equivalent.